Being Easier to Win Doesn't Always Make a Bet more Advantageous

It's usually true.

Here's an example. The chance of winning a "straight-up" bet on a number at single-zero roulette is one out of 37, 2.70 percent. The probability of winning the same proposition in a double-zero game is one out of 38, 2.63 percent. Payoffs are 35-to-1 either way. The single-zero wager is more likely to win, 2.70 versus 2.63 percent. And the probability increment is more than matched where it counts. Player expectation is to lose $2.70 per $100 at risk on the single-zero bet as opposed to $5.26 on the double.

As another illustration, consider betting $10 at craps, either on the four or the five. A win on the four pays $18 and has a chance of one out of three or 33.33 percent. Success on the five pays $14 has a probability of two out of five or 40.00 percent. So a shot at the five is easier to win, 40.00 versus 33.33 percent. And, despite its lower payoff, the five also yields less of an expected average loss: $4.00 as opposed to $6.67 per $100 bet.

"Usually," however, isn't "always." A gamble that's easier to win may not be more profitable or less costly than some alternative.

One reason for the reversal is that an imbalance in payoffs might make a tougher bet statistically more advantageous. For instance, the chance of winning on a hard four at craps is one out of nine or 11.11 percent; the payoff is 7-to-1. The probability of winning on a hard six is one out of 11 or 9.09 percent; the payoff is 9-to-1. The four is easier to win, 11.11 versus 9.09 percent. However, expected loss per $100 bet works oppositely. It's $9.09 on the hard six; it's worse -- $11.11 -- on the four.

Pushes are a second, more subtle, reason why a better chance of winning may be less advantageous than the alternative. The pesky 12 versus two-up at blackjack illustrates this phenomenon. The hand is an underdog no matter how it's executed. Basic Strategy is to hit, but not everybody is comfortable doing so. And the flip-flop between chances of winning and expectation may partly account for solid citizens daring to doubt the dogma.

A player who stands on 12 will win if the dealer busts and will lose otherwise. No pushes are possible. The chances are 35.36 percent of winning and the remaining 64.64 percent of losing. For every $100 up for grabs when this hand occurs, players expect to win $35.36 and lose $64.64. The net effect is an average loss of $64.64 - $35.36 or $29.28 per $100 bet.

The chance of players winning by hitting is more complicated. It's that of finishing at or under 17 and the dealer busting, or ending at 18 and the dealer getting 17 or busting, and so on up to sitting at 21 and the dealer getting 17 through 20 or busting. The arithmetic for a win comes out to 34.84 percent. Conversely, the chance of losing is that of busting, or looking at a total under 17 and the dealer getting 17 through 21, or of finishing at 17 and the dealer getting 18 through 21, and so on up to scoring a 20 and the dealer reaching 21. The bottom line for a loss is 60.18 percent. The chance of a push -- of finishing at 17 and the dealer doing likewise, or at 18 and the dealer matching this total, up through both ending at 21 -- is 4.97 percent.

Per $100 bet, this averages $34.84 in wins, $60.18 in losses, and $4.97 in pushes. The net is $60.18 - 34.84 or a $25.34 whack.

Players will win more often on 12 versus two by standing than hitting, 35.36 versus 34.84 percent. But, pushes when players stand erode the probability of a loss more than that of a win, so the expected penalty associated with the hand is less with a hit, $25.34 in contrast with $29.28 per $100 on the layout. And Basic Strategy is predicated on expected gains or losses on a hand, not on the chances of triumph or tragedy. Which goes to prove the prescience of the poet, Sumner A Ingmark, when he penned:

How you view the circumstances,
Impacts how you view your chances.